A parametric model to jointly characterize rate, duration, and severity of exacerbations in episodic diseases

Background The natural history of many chronic diseases is characterized by periods of increased disease activity, commonly referred to as flare-ups or exacerbations. Accurate characterization of the burden of these exacerbations is an important research objective. Methods The purpose of this work was to develop a statistical framework for nuanced characterization of the three main features of exacerbations: their rate, duration, and severity, with interrelationships among these features being a particular focus. We jointly specified a zero-inflated accelerated failure time regression model for the rate, an accelerated failure time regression model for the duration, and a logistic regression model for the severity of exacerbations. Random effects were incorporated into each component to capture heterogeneity beyond the variability attributable to observed characteristics, and to describe the interrelationships among these components. Results We used pooled data from two clinical trials in asthma as an exemplary application to illustrate the utility of the joint modeling approach. The model fit clearly indicated the presence of heterogeneity in all three components. A novel finding was that the new therapy reduced not just the rate but also the duration of exacerbations, but did not have a significant impact on their severity. After controlling for covariates, exacerbations among more frequent exacerbators tended to be shorter and less likely to be severe. Conclusions We conclude that a joint modeling framework, programmable in available software, can provide novel insights about how the rate, duration, and severity of episodic events interrelate, and enables consistent inference on the effect of treatments on different disease outcomes. Trial registration Ethics approval was obtained from the University of British Columbia Human Ethics Board (H17-00938). Supplementary Information The online version contains supplementary material available at 10.1186/s12911-022-02080-5.


Collapsing treatment arms across studies
As a preliminary step, we checked the comparability of the covariates across studies to see whether it is reasonable to collapse the data on the two treatment arms that are common to the two studies (placebo and 75 mg Mepolizumab). Table 1 suggests some differences so we proceeded to more formal assessment via model comparisons, incorporating study ID and its interactions with the common treatment arms as extra predictors in all the submodels. These extra predictors had little impact on the model goodness of fit, as assessed both by model AIC and significance level (p-value> 0.1 for all these extra predictors).
The results indicate that collapsing common treatment arms across studies is reasonable. We begin by describing the contribution of patient i to the likelihood conditional on the random effects for our case study. One can modify these contributions to the settings of other applications in a straightforward fashion. If patient i has no exacerbations over the follow-up period, its contribution to the conditional likelihood is

Supplementary
For patients with at least one exacerbation over the follow-up period, let a i = 1 if patient i ends the nominal follow-up period during an exacerbation (in this case, T i = v i,Mi , the time of the end of this exacerbation) and a i = 0 otherwise. The contribution of patient i to the conditional likelihood is Note that if the non-susceptible component is not required in an application, one can set π i ≡ 0.
To obtain the full likelihood, we integrate the contribution of patient i over the multivariate normal distribution of the three random effects and take the product of these (marginal) likelihood contributions across all patients. Table 2 shows the AIC results of four different baseline hazard functions when these are used for both the rate and duration components of the model (the AIC results were similar for other combinations of these distributions for the rate and duration submodels). The log-normal distribution for both rate and duration submodels has the lowest AIC for the case study.

AIC comparison
Supplementary

Correlation between outcomes
We can use the estimated random effect values (along with other estimated regression coefficients) to predict all the outcomes (between and within exacerbation times, and severity statuses) for each patient. We can then use these predicted values to estimate the correlations between each pair of outcomes based on our fitted model. The average of such correlations over multiple predicted Supplementary Figure 1. Correlation heatmap plot of the first three episodes among the predictions (left) and observations (right) data sets estimates the true correlation. We generated 100 predicted data sets for patients who had at least three exacerbations during the follow-up time. Figure 1 shows the heatmap plots for the correlations between the predicted (left panel) and observed outcomes (right panel) for the first three episodes. We use the Pearson correlation coefficients for pairs with numeric outcomes (between and within exacerbation times) and Kendall's tau coefficient (association) otherwise. The pattern of the two sets of correlations are similar except the magnitude of correlations in the predictions are slightly larger than those of the observations. The results indicate that the assumed correlation structure in the model is reasonable.

Sensitivity Analysis
To assess the impact of excluding patients with missing values in their covariates, we carried out a sensitivity analysis. We used multiple imputation to generate 5 different datasets with imputed values for the missing values, and interpreted the treatment effect (main covariates of interest) as well as the random effect parameter estimates in terms of their sensitivity to the use of complete cases versus imputed data. As the Table 3 reports, the estimates from the two analyses were. Supplementary